High-low method and $p$-adic Furstenberg set over the plane
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909883150893056 |
|---|---|
| author | Ren, Kevin Shen, Jiahe |
| author_facet | Ren, Kevin Shen, Jiahe |
| contents | We establish a $p$-adic analogue of a recent significant result of Ren-Wang (arXiv:2308.08819) on Furstenberg sets in the Euclidean plane. Building on the $p$-adic version of the high-low method from Chu (arXiv:2510.20104), we analyze cube-tube incidences in $\mathbb{Q}_p^2$ and prove that for $s < t < 2 - s$, any semi-well-spaced $(s,t)$-Furstenberg set over $\mathbb{Q}_p^2$ has Hausdorff dimension $\ge\frac{3s+t}{2}$. Moreover, as a byproduct of our argument, we obtain the sharp lower bounds $s+t$ (for $0<t\le s\le 1$) and $s+1$ (for $s+t\ge 2$) for general $(s,t)$-Furstenberg sets without the semi-well-spaced assumption, thereby confirming that all three lower bounds match those in the Euclidean case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_01257 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | High-low method and $p$-adic Furstenberg set over the plane Ren, Kevin Shen, Jiahe Functional Analysis 28A78 (primary), 28A80 (secondary) We establish a $p$-adic analogue of a recent significant result of Ren-Wang (arXiv:2308.08819) on Furstenberg sets in the Euclidean plane. Building on the $p$-adic version of the high-low method from Chu (arXiv:2510.20104), we analyze cube-tube incidences in $\mathbb{Q}_p^2$ and prove that for $s < t < 2 - s$, any semi-well-spaced $(s,t)$-Furstenberg set over $\mathbb{Q}_p^2$ has Hausdorff dimension $\ge\frac{3s+t}{2}$. Moreover, as a byproduct of our argument, we obtain the sharp lower bounds $s+t$ (for $0<t\le s\le 1$) and $s+1$ (for $s+t\ge 2$) for general $(s,t)$-Furstenberg sets without the semi-well-spaced assumption, thereby confirming that all three lower bounds match those in the Euclidean case. |
| title | High-low method and $p$-adic Furstenberg set over the plane |
| topic | Functional Analysis 28A78 (primary), 28A80 (secondary) |
| url | https://arxiv.org/abs/2511.01257 |